I am looking for some metrics/analysis comparing direct mail vs voice messaging

voice messaging can also be referred to as voice broadcasting and
voice marketing

it is a technology used to leave voice messages on home answering
machines vs leaving a message in a live ear

I can give you an analysis from the standpoint of the recipient.

Direct mail ads can be tossed in the trash. They are moderately
annoying.

On the other hand, ads that appear on my answering machine are using
valuable storage space that I would prefer be allotted to actual
messages from the people I care about. If I find that my answering
machine is filled with spam, and this has kept someone from leaving me
a legitimate message, you can well imagine my attitude toward the
originator of the spam. Not only will I never purchase anything from
someone who advertises in this way, but I will urge others to boycott
companies who use this form of advertising.

Hear has what has been successful for me:

Send out a direct mail piece, and then follow it
up with a voice broadcast message.

You can also experiment, and try sending the
voice mail message out first.

This 1-2 punch works amazingly well.